Commun. Theor. Phys. Beijing, China) 54 2010) pp. 797 802 c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 5, November 15, 2010 Approximate Similarity Reduction for Perturbed Kaup Kupershmidt Equation via Lie Symmetry Method and Direct Method LIU Xi-Zhong ) Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China Received March 3, 2010) Abstract The perturbed Kaup Kupershmidt equation is investigated in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions of different orders are obtained for both methods, series reduction solutions are consequently derived. Higher order similarity reduction equations are linear variable coefficients ordinary differential equations. By comparison, it is find that the results generated from the approximate direct method are more general than the results generated from the approximate symmetry perturbation method. PACS numbers: 02.30.Jr, 04.20.Jb, 02.20.Sv, 02.30.Ik Key words: perturbed Kaup Kupershmidt equation, approximate symmetry perturbation method, approximate direct reduction method, series reduction solutions 1 Introduction It is a fundamental task and of great importance to solve nonlinear partial differential equations. Lie group theory, [1 5] which was originally created more than 100 years ago is now a very general and useful tool for obtaining exact analytic solutions of large classes of differential equations, specially nonlinear ones. Using the classical Lie group theory, a system of over determined linear equations are treated. Once the symmetries of the equations are obtained, similarity solutions can be produced. There have been several generalizations of the classical Lie group method for symmetry reduction, one of which is the notion of nonclassical symmetries due to Bluman and Cole. [6 8] Although nonclassical symmetries can yield more solutions than the classical symmetries method, it is difficult to solve because the corresponding determining equations are no longer linear. Besides these two methods, Clarkson and Kruska [9] proposed the direct method, which does not use techniques of group analysis and obtained previously unknown symmetry reductions for the Boussinesq equation. Many PDEs in applications depend on a small parameter, so it is of great importance and interest to find approximate solutions. The perturbation analysis method which has been widely applied to nonlinear problems is used to obtain approximate analytical solutions by expanding the dependent variables asymptotically in terms of a small parameter. The combination of Lie Group Theory and perturbations results in two different approximate symmetry theories. In the first method which was first introduced by Baikov, Gazizov and Ibragimov, [10 11] the infinitesimal generator is expanded in a perturbation series other than perturbation for dependent variables as in the usual case. In the second method i.e. the approximate symmetry method) due to Fushchich and Shtelen [12] the dependent variables are expanded in a perturbation series first, terms are then separated at each order of approximation and a system of equations to be solved in a hierarchy is obtained. The approximate symmetry of the original equation is defined to be the exact symmetry of the system of equations obtained from perturbations. Apart from the approximate symmetry techniques, another interesting approximate method using direct method has been proposed in Ref. [13] and applied in many problems recently. The well-known Kaup Kupershmidt KK) equation u t + 180u 2 u x + 75u x u xx + 30uu xxx + u xxxxx = 0, 1) which was first introduced by Kaup [14] in 1980 has been widely studied over the past two decades. Because of the difficulty to analyze KK equation 1) by standard means, its N-soliton solutions have been found only in the last four years [15 21] and other classes of exact solutions for Eq. 1) have been considered in Refs. [22 23]. There are many kinds of perturbations for the KK equation, in this paper, we consider this equation with a perturbation of the following form u t + 180u 2 u x + 75u x u xx + 30uu xxx + u xxxxx + ǫαu 3 u x uu x u xx ) = 0, 2) where ǫ is a small parameter, α and γ are arbitrary constants. Following the perturbation theory, the solution for Eq. 2) can be expanded in a perturbation series u = ǫ j u j, 3) j=0 Supported by the National Natural Science Foundation of China under Grant Nos. 10735030, 10475055, 10675065, and 90503006, National Basic Research Program of China 973 Program 2007CB814800)
798 LIU Xi-Zhong Vol. 54 where u j are functions of x and t. Substituting Eq. 3) into Eq. 2) and vanishing the coefficients of all different powers of ǫ, we get a system of partial differential equations j u k,t + 180 u l,x u j l u k j + 75 u i,xx u k i,x + 30 j u i,x u j i u j,xx l l=0 j u i,x u j i u l j u l + u k,xxxxx u i,xxx u k i = 0, k = 0, 1,...), 4) with u 1 = 0. The paper is organized as follows. In Secs. 2 and 3, we apply the symmetry reduction method and direct method to Eq. 4), respectively, to obtain similarity reduction solutions and similarity reduction equations of different orders. The last section is a conclusion and discussion of the results. 2 Approximate Symmetry Reduction Approach to Eq. 2) In order to get exact solutions of Eq. 4), we first construct its Lie point symmetries and then give the corresponding symmetry reductions. A symmetry is characterized by an infinitesimal transformation which leaves the given differential equation invariant under the transformation of all independent and dependent variables. We assume the corresponding Lie point symmetry has the vector form V = X x + T t + U k, u k k=0 where X, T, and U k are functions with respect to x, t, and u k, k = 0, 1,..., which means that the system of Eq. 4) is invariant under the following transformations {x, t, u k, k = 0, 1,...} {x + εx, t + εt, u k + εu k, k = 0, 1,...}, 5) with a small parameter ǫ. Or, we can write the symmetry in the function form σ k = U k Xu k,x Tu k,t, k = 0, 1, 2,... 6) σ k is the solution of the linearized equations for Eqs. 4) σ k,t + 180 l l=0 + 30 j σ l,x u j l u k j + u l,x σ j l u k j + u l,x u j l σ k j ) + 75 σ i,xx u k i,x + u i,xx σ k i,x ) j σ i,x u j i u l j u l + u i,x σ j i u l j u l + u i,x u j i σ l j u l + u i,x u j i u l j σ l ) j σ i,x u j i u j,xx + u i,x σ j i u j,xx + u i,x u j i σ j,xx ) + σ k,xxxxx σ i,xxx u k i + u i,xxx σ k i ) = 0, k = 0, 1, 2,...), 7) with σ 1 = 0. Substituting Eq. 6) into the linearized equations 7) and eliminating u k,t in terms of Eq. 4), we obtain the determining equations for the functions X, T, and U k k = 0, 1,...). The solutions to the determining equations are X = C 1 5 x + C 3, T = C 1 t + C 2, U 0 = 2C 1 5 u 0, U 1 = 4C 1 5 u 1, U 2 = 6C 1 5 u 2,..., U k = 2k + 2 C 1 u k, k = 0, 1, 2,...), 8) 5 where C 1, C 2, and C 3 are arbitrary constants. Solving the characteristic equations dx X = dt T, du 0 = dt U 0 T, du 1 = dt U 1 T,..., du k U k = dt T,..., 9) similarity reductions can be derived, which will be discussed in detail for the following two cases. Case 1 When C 1 0, for simplicity, we make the transformations C 2 C 1 C 2 and C 3 1/5)C 1 C 3, so Eq. 8) becomes X = 1 5 C 1x + C 3 ), T = C 1 t + C 2 ), U k = 2k + 2 C 1 u k, k = 0, 1,...). 10) 5 We can get the similarity variable from Eqs. 9) and 10). ξ = x + C 3 5 t + C2. 11)
No. 5 Approximate Similarity Reduction for Perturbed Kaup Kupershmidt Equation via Lie Symmetry Method 799 Then the similarity solutions for the u k are u k = V k ξ), k = 0, 1,...). 12) t + C 2 )2k+2)/5 Accordingly, the perturbation series solution of Eq. 2) is of the form u = k=0 ǫ k V k ξ), 13) t + C 2 )2k+2)/5 with invariant functions V k satisfy the following reduction equations 5V k,ξξξξξ + 900 + 5γ j V l,ξ V j l V k j + 5α j V i,ξ V j i V j,ξξ + 375 l l=0 j V i,ξ V j i V l j V l + 150 V i,ξξξ V k i V i,ξξ V k i,ξ ξv k,ξ 2k + 2)V k = 0, k = 0, 1,...), 14) with V 1 = 0. When taking k = 0, after integrating and choosing integral constant as 0, Eq. 14) can be solved by the WeierstrassP i.e. W in Eq. 15)) function V 0 = 4m 2 1W m 2 + m 1 ξ, ξ ), 0, 15) 330m 4 1 with m 1, m 2 arbitrary constants. When k > 0, from Eq. 14), we see that the k-th similarity reduction equation is a fifth order linear ordinary differential equation of V k when the previous V 0, V 1,..., V are known. In fact, we can theoretically compute V 0, V 1,..., V k one after another. To see this point we rewrite Eq. 14) as 5V k,ξξξξξ + 1800V 0ξ V 0 V k + 900V k,ξ V 2 0 ξv k,ξ 2k + 2)V k + 150V 0,ξξξ V k + 150V k,ξξξ V 0 +375V 0,ξξ V k,ξ + 375V k,ξξ V 0,ξ = G k, 16) where G k is only a function of {V 0, V 1,...,V }, which is defined by j G k 900 V l,ξ V j l V k j + 900 V l,ξ V k l V 0 j=1 l=0 + 5α + 5γ l l=0 j V i,ξ V j i V l j V l j V i,ξ V j i V j,ξξ + 375 V i,ξξ V k i,ξ + 150 V i,ξξξ V k i. Case 2 When C 1 = 0, solving Eq. 9) we can get the following similarity solutions u k = V k ξ), k = 0, 1,...), 17) with the similarity variable ξ = t C 2 /C 3 )x, which can be redefined as ξ = x + ct with c an arbitrary velocity constant. So the the series reduction solution related to Eq. 17) is u = ǫ k V k ξ). 18) k=0 We can get the following reduction equations for V k ξ) by substituting Eq. 17) into Eq. 4) V k,ξξξξξ + 180 j V l,ξ V j l V k j j V i,ξ V j i V j,ξξ + 75 l l=0 j V i,ξ V j i V l j V l + 30 V i,ξξξ V k i V i,ξξ V k i,ξ + cv k,ξ = 0, k = 0, 1,...), 19) with V 1 = 0. When k = 0, the general solution of Eq. 17) can be expressed by the Weierstrass function V 0 = 4h 2 2W h 3 + h 2 ξ, c ) 132h 4, h 1, 20) 2 where W is the Weierstrass P function, h 1, h 2, and h 3 are arbitrary constants. The n-th similarity reduction equation in Eq. 19) is a fifth order linear ordinary differential equation of V n when the previous V 0, V 1,..., V n 1 are known. Similar to Eq. 14), Eq. 19) can be solved one after another through the following equation V k,ξξξξξ + 360V 0,ξ V 0 V k + 180V k,ξ V 2 0 + 30V 0,ξξξV k +30V k,ξξξ V 0 +75V 0,ξξ V k,ξ +75V k,ξξ V 0,ξ +cv k,ξ = H k, 21) where H k is given by H k 180 j j=1 l=0 l l=0 V l,ξ V j l V k j + 180 V l,ξ V k l V 0 j V i,ξ V j i V l j V l
800 LIU Xi-Zhong Vol. 54 j V i,ξ V j i V j,ξξ + 75 V i,ξξ V k i,ξ + 30 V i,ξξξ V k i. 3 Approximate Direct Method to Eq. 2) The direct method proposed by CK consists in finding the similarity reduction of Eq. 4) in the form u j = U j x, t, P j zx, t))), j = 0, 1,...), 22) where U j, P j, and z are functions with respect to the indicated variables and P j z) satisfy a system of ordinary differential equations, which can be obtained by substituting Eq. 22) into Eq. 4). In fact, we can choose a special form of Eq. 22) for Eq. 4), namely u j = α j x, t) + β j x, t)p j zx, t)), j = 0, 1,...). 23) To see this point, we substitute Eq. 22) into only one term u j,xxx in Eq. 4), since only this term generates P j,zzz and P j,zz P j,z after the substitution. It is easily seen that the coefficients for P j,zzz and P j,zz P j,z are U j,pj zx 5 and 5U j,pjp j zx 5 respectively. We reserve uppercase Greek letters for undetermined functions of z hereafter. Because it must be an ordinary differential equation in P j z) after substituting Eq. 22) into Eq. 4), the ratios of the coefficients are functions of z, namely, 5U j,pjp j z 5 x = U j,p j z 5 x Γ jz), j = 0, 1,...), 24) with the solution U j = F j x, t) + G j x, t)e 1/5)Γjz)Pj. So, it is sufficient to seek similarity reductions of Eqs. 4) in the form 23). Remark Without loss of generality, there are three freedoms in the determination of α j x, t), β j x, t), and zx, t): i) If α j x, t) has the form α j x, t) = α j x, t)+β jωz), then one can take Ωz) = 0; ii) If β j x, t) has the form β j x, t) = β j x, t)ωz), then one can take Ωz) = constant; iii) If zx, t) is determined by Ωz) = z 0 x, t), where Ωz) is any invertible function, then one can take Ωz) = z. We substitute Eq. 23) into Eq. 4) of k = 0 case and single out the coefficients for P 0zzzzz, P 0zzz P 0, P 0zzz, and P 0zzzz, which are β 0 zx 5, 30β2 0 z3 x, 30α 0β 0 zx 3 + 30β 0xzx 2z xx + 10β 0 zxz 2 xxx + 15β 0 z x zxx 2 + 10β 0xx zx, 3 and 10β 0 zxz 3 xx + 5β 0x zx 4, respectively. The assumption P j is only a function of z requires that 30β 2 0z 3 x = β 0 z 5 xψ 0 z), 25) 30α 0 β 0 z 3 x + 30β 0xz 2 x z xx + 10β 0 z 2 x z xxx + 15β 0 z x z 2 xx + 10β 0xx z 3 x = β 0 z 5 xφ 0 z), 26) 10β 0 z 3 xz xx + 5β 0x z 4 x = β 0 z 5 xω 0 z), 27) and thus, from Eqs. 25) and Remark ii), we have β 0 = z 2 x. 28) From Eqs. 26), 28), and Remark i), we have α 0 = 19z2 xx 6z 2 x z xxx z x. 29) From Eqs. 27), 28), and Remark iii), after redefining z we have z = θt)x + σt), 30) where θt) and σt) are undetermined functions of integration. Then Eq. 4) of k = 0 case is simplified to 180θ 7 P 2 0 P 0z + 2θ t θ + 30θ 7 P 0zzz )P 0 + 75θ 7 P 0zz +θθσ t + θ t z θ t σ))p 0z + θ 7 P 0zzzzz = 0. 31) For this to be an ordinary differential equation in P 0 z), the coefficients of P 0zzz, zp 0z, and P 0z must satisfy the following relations θ t = A 1 θ 6, θσ t θ t σ = A 2 θ 6, 32) with A 1 and A 2 being arbitrary constants. Case 1 When A 1 0, Eq. 32) has the solution θ = 5A 1 t t 1 ) 1/5, σ = A 2 A 1 + t 2 5A 1 t t 1 ) 1/5, 33) where t 1 and t 2 are arbitrary constants. In order to determine α j, β j, we substitute Eq. 23) into Eq. 4) and single out the coefficients of P j,zzzzz, P 3 0 P j,z, and P 0zzz, which are β j z 5 x, αβ3 0 β j 1z x, and 30α j β 0 z 3 x, respectively. We require that αβ 3 0 β j 1z x = β j z 5 x Ψ jz), j = 1, 2,...), 34) 30α j β 0 z 3 x = β jz 5 x Φ jz), j = 1, 2,...). 35) From Eqs. 28), 34), and Remark ii), we obtain β j = z 2j+2 x, j = 0, 1,...). 36) From Eqs. 28), 35), 36) and Remark i), we obtain α j = 0, j = 1,...). 37) In terms of Eqs. 3), 23), 30), 33), 36), and 37), we get the similarity solutions to Eq. 4) u j = 5A 1 t t 1 ) 1/5)2j+2) P j z), 38) with the similarity variable z = 5A 1 t t 1 ) 1/5 x + t 2 5A 1 t t 1 ) 1/5 A 2. A 1 So the series reduction solutions to Eq. 2) can be derived from Eqs. 3) and 38) u = 5A 1 t t 1 ) 1/5)2j+2) ǫ j P j z). 39) j=0 The related similarity reduction equations are
No. 5 Approximate Similarity Reduction for Perturbed Kaup Kupershmidt Equation via Lie Symmetry Method 801 P j,zzzzz + 180 j 1 +γ k=0 j k=0 l=0 j 1 P l,z P k l P j k + 75 l l=0 k=0 + 30 j P i,zzz P j i j P i,zz P j i,z + za 1 + A 2 )P z + 2k + 2)A 1 P j = 0, k = 0, 1,...), 40) with P 1 = 0. We now compare Eqs. 38) and 40) with Eqs. 12) and 14). Making the transformation t 2 C 3, A 2 0, A 1 1/5, and t 1 C 2 simultaneously, we see that the similarity variable becomes z = x + C 3 )/ 5 t + C 2 ; Eqs. 38) and 40) are changed into P j z) u j =, j = 0, 1,...), 41) t + C 2 )2j+2)/5 j 5P j,zzzzz + 900 P l,z P k l P j k j 1 + 5α + 150 + 375 l l=0 k=0 j k=0 l=0 j 1 P i,zzz P j i + 5γ k=0 j P i,zz P j i,z zp z 2k + 2)P j = 0, j = 0, 1,...), 42) with P 1 = 0. Obviously, Eqs. 41) and 42) are exactly the same as Eqs. 12) and 14). When j = 0, after integrating and taking integral constant as 0, Eq. 40) can be solved by P 0 = 4e 2 4W e 3 + e 4 z, A 2 + 2zA 1 132e 4 4 ), 0, 43) where W is the WeierstrassP function and e 3, e 4 are arbitrary constants. Equation 40) is in fact a fifth order linear variable coefficients ordinary differential equation of P j with the assumption that P k for k < j are determined in advance, since we can rewrite Eq. 40) as P j,zzzzz + 360P 0,z P 0 P j + 180P j,z P 2 0 + 30P 0,zzz P j +30P j,zzz P 0 + 75P 0,zz P j,z + 75P j,zz P 0,z + za 1 + A 2 )P z +2k + 2)A 1 P j = f j, j = 0, 1,...), 44) where j 1 f j = 180 k=1 l=0 j 1 l l=0 k=0 j 1 P l,z P k l P j k + 180 P l,z P j l P 0 j 1 j 1 + 30 P i,zzz P j i j 1 + 75 P i,zz P j i,z. k=0 Case 2 When A 1 = 0, it is straightforward from Eq. 32) to find that θ = θ 0, σ = A 2 θ 5 0 t + σ 0, 45) with θ 0 and σ 0 arbitrary constants. From Eqs. 30) and 45), we can redefine the similarity variable as z = x + ct which has the travelling wave form. Thus we obtain the perturbation series solution to Eq. 2) u = ǫ j P j z), 46) with P j z) satisfy j P j,zzzzz + 180 j 1 +α +30 +75 l l=0 k=0 k=0 l=0 j=0 P l,z P k l P j k j j 1 P i,zzz P j i k=0 j P i,zz P j i,z + cp j,z = 0, j = 0, 1,...), 47) where P 1 = 0. Obviously, Eq. 47) is exactly the same as Eq. 19). When taking j = 0 for Eq. 47), we find that the solution can be given by P 0 = 4c 2 2W c 3 + c 2 z, c ) 132c 4, c 4, 48) 2 where W is WeierstrassP function, c 2, c 3, c 4 are arbitrary constants. When j > 0, similar to Eq. 44), we can obtain P j j = 1, 2,...) through the following recursion equation P j,zzzzz + 360P 0,z P 0 P j + 180P j,z P 2 0 + 30P j,zzzp 0 + 30P 0,zzz P j + 75P 0,zz P j,z + 75P j,zz P 0,z + cp j,z = g j, j = 0, 1,...), 49) where j 1 j 1 g j = 180 P l,z P k l P j k + 180 P l,z P j l P 0 k=1 l=0
802 LIU Xi-Zhong Vol. 54 j 1 l l=0 k=0 j 1 j 1 + 30 P i,zzz P j i j 1 + 75 P i,zz P j i,z. 4 Conclusion Remarks k=0 In summary, approximate similarity reduction of the perturbed Kaup Kupershmidt equation was studied in terms of the approximate symmetry method and the approximate direct method. We summarized the general formulas for the similarity reduction solutions and similarity reduction equations of different orders for both methods. The series reduction solutions were consequently derived. Similarity reduction equations of zero order obtained from both approximate symmetry perturbation method and approximate direct method can be expressed by WeierstrassP equations. P j can be solved out step by step through the corresponding similarity reduction equation which is linear in P j when the previous P 0 to P j 1 are known. By comparison, it is discovered that the similarity reductions generated from approximate direct method has more general form than the corresponding results generated from the approximate symmetry method. References [1] G.W. Bluman and S. Kumei, Symmetries and Differential Equations, Applied Mathematical Sciences Vol. 81, Springer-Verlag, Berlin 1989). [2] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York 1993). [3] N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Nauka, Moscow 1983). [4] L.V. Ovsiannikov, Group Analysis of Differential Equations, Nauka, Moscow 1978). [5] H. Stephani, Differential Equations: Their Solutions Using Symmetries, Cambridge University Press, Cambridge 1989). [6] G.W. Bluman and J.D. Cole, J. Math. Mech. 18 1969) 1025. [7] W.F. Ames, Nonlinear Partial Differential Equations in Engineering: II, Academic, New York 1972). [8] B.K. Harrison and F.B. Estabrook, J. Math. Phys. 12 1971) 653. [9] P.A. Clarkson and M.D. Kruskal, J. Math. Phys. 30 1989) 2201. [10] V.A. Baikov, R.K. Gazizov, and N.H. Ibragimov, Mat. Sb. 136 1988) 435 [Math. USSR. Sb. 64 1989) 427]. [11] V.A. Baikov, R.K. Gazizov, and N.H. Ibragimov, in Approximate Transformation Groups and Deformations of Symmetry Lie Algebras, CRC Handbook of Lie Group Analysis of Differential Equations Vol. 3, ed. by N.H. Ibragimov, CRC, Boca Raton, FL 1996). [12] W.I. Fushchich and W.M. Shtelen, J. Phys. A: Math. Gen. 22 1989) 887. [13] X.Y. Jiao, R.X. Yao, and S.Y. Lou, J. Math. Phys. 49 2008) 093505. [14] D.J. Kaup, Stud. Appl. Math. 62 189 1980). [15] W. Hereman and A. Nuseir, Math. Comput. Simul. 43 1997) 13. [16] A. Parker, Physica D 137 2000) 25. [17] C. Verhoeven and M. Musette, J. Phys. A 34 2001) 2515. [18] M. Musette and C. Verhoeven, Physica D 144 2000) 211. [19] M. Musette and R. Conte, J. Math. Phys. 39 1998) 5617. [20] P. Han and S.Y. Lou, Acta Phys. Sin. 43 1994) 1042. [21] S.Y. Lou and H.Y. Ruan, J. Phys. A: Math. Gen. 26 1993) 4679. [22] R.A. Zait, Chaos, Solitons and Fractals 15 2003) 673. [23] W.P. Hong and Y.D. Jung, Z. Naturforsch. A: Phys. Sci. 54 1999) 549.